Civil Engineering Reference
In-Depth Information
The “implicit” strategies described above are quite effective for linear problems (con-
stant [
K
c
] and [
M
m
]), however storage requirements can be considerable, and in non-linear
problems the necessity to refactorise
(
[
M
m
]
+
θt
[
K
c
]
)
can lead to lengthy calculations.
Storage can be saved by replacing subroutine
linmul_sky
by an element-by-element
matrix-vector multiply, and by solving the simultaneous equations iteratively (e.g. by
pcg). Such a strategy can be attractive for parallel processing and examples are given
in Chapter 12.
There is another alternative, widely used for second-order problems (see also
Section 3.13.4), in which
θ
is set to zero and the [
M
m
] matrix is “lumped” (see
equation 3.67). In this “explicit” approach, the system to be solved is
[
M
m
]
{
}
1
=
(
[
M
m
]
−
t
[
K
c
]
)
{
}
0
(3.97)
or
[
M
m
]
−
1
(
[
M
m
]
{
}
1
=
−
t
[
K
c
]
)
{
}
0
(3.98)
Although written here at the global level, in the case of
θ
=
0noglobalmatrixassembly
is needed because the matrix-vector products on the right hand side of equation (3.98)
can all be achieved using an element-by-element strategy involving manipulations of the
element matrices [
k
c
] and [
m
m
].
Although the “explicit” algorithm is simple, the disadvantage is that (3.98) is only stable
on condition that
t
is “small”, and in practice perhaps so small that real times of interest
would require an excessive number of steps.
Yet another element-by-element approach which conserves computer storage while pre-
serving the stability properties of “implicit methods” involves “operator splitting” on an
element-by-element product basis (Hughes
et al
., 1983; Smith
et al
., 1989). Although not
necessary for the operation of the method, the simplest algorithms result from “lumping”
[
M
m
]. Assuming again that
{
Q
} = {
0
}
, equation (3.94) can be written,
+
θt
[
K
c
]
)
−
1
(
[
M
m
]
{
}
1
=
(
[
M
m
]
−
(
1
−
θ)t
[
K
c
]
)
{
}
0
(3.99)
The element-by-element “operator splitting” methods are based on binomial theorem expan-
sions of
(
[
M
m
]
+
θt
[
K
c
]
)
−
1
which neglect product terms. When [
M
m
] is “lumped”
(diagonal), the method is particularly straightforward because [
M
m
] can effectively be
replaced by [
I
], the unit matrix, where
[
I
]
+
θt
[
k
c
]
−
1
+
θt
[
K
c
]
)
−
1
(
[
I
]
=
(3.100)
(
[
I
]
+
θt
[
k
c
]
)
−
1
(3.101)
where
indicates a summation, and
indicates a product over all the elements. As
was the case with implicit methods, optimal accuracy consistent with stability is achieved
for
θ
=
≈
1
/
2. It is shown by Hughes
et al
. (1983) that further optimisation is achieved by
splitting further to
1
2
[
k
c
]
1
2
[
k
c
]
[
k
c
]
=
+
(3.102)
